Problem: $ D = \left[\begin{array}{rrr}5 & 5 & -2 \\ 1 & -1 & 5\end{array}\right]$ $ E = \left[\begin{array}{rr}3 & 2 \\ -2 & 3 \\ 1 & 1\end{array}\right]$ What is $ D E$ ?
Solution: Because $ D$ has dimensions $(2\times3)$ and $ E$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ D E = \left[\begin{array}{rrr}{5} & {5} & {-2} \\ {1} & {-1} & {5}\end{array}\right] \left[\begin{array}{rr}{3} & \color{#DF0030}{2} \\ {-2} & \color{#DF0030}{3} \\ {1} & \color{#DF0030}{1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{5}\cdot{3}+{5}\cdot{-2}+{-2}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{3}+{5}\cdot{-2}+{-2}\cdot{1} & ? \\ {1}\cdot{3}+{-1}\cdot{-2}+{5}\cdot{1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{3}+{5}\cdot{-2}+{-2}\cdot{1} & {5}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{3}+{-2}\cdot\color{#DF0030}{1} \\ {1}\cdot{3}+{-1}\cdot{-2}+{5}\cdot{1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{5}\cdot{3}+{5}\cdot{-2}+{-2}\cdot{1} & {5}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{3}+{-2}\cdot\color{#DF0030}{1} \\ {1}\cdot{3}+{-1}\cdot{-2}+{5}\cdot{1} & {1}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}3 & 23 \\ 10 & 4\end{array}\right] $